A model of plant-herbivore interaction with size structure in plants using toxin-determined functional response

One of the possible consequences of global warming in the northern hemisphere is the invasion of Arctic Tundra by woody plants, which is influenced by the chemical defense of plants against mammal browsing. In this paper, we explore the toxin-mediated plant-herbivore interaction with size structure in plants using a simplified mathematical model. The size structure of plants is important as it seems to play a significant role on the woody plant invading. We formulate a three dimensional ordinary differential equations model of the simplified dynamics of adult-trees, seedlings and herbivores in the presence of chemical defense of plants, building on a general two dimensional toxin-determined functional response model of plants and herbivores. Then we investigate the mathematical features of the model, analyze the stability of equilbiria. When choosing plant toxin measurement G and herbivore health measurement μ as bifurcation parameters, we analytically show the existence of different bifurcations, including a saddle-node bifurcation and a Hopf bifurcation. A BogdanovTakens cusp with codimension 2 is also identified. A Bogdanov-Takens bifurcation occurs when bifurcation parameters are perturbed at the cusp. ∗email: [email protected] †email: [email protected]